Speaker
Description
Inverse problems on a graph investigate how spectral behaviors interact with the matrices associated with the given graph. Such problems not only uncover structural information about the graph from its spectral data, but also identify fundamental properties shared by all matrices defined on the graph. A classic example is the Colin de Verdière parameter, which characterizes planarity via the maximum nullity over matrices defined on the graph. Central to these studies are strong properties, which either preserve the matrix pattern while perturbing spectral data or preserve the spectral data while adjusting the pattern. A recurring theme is that if a spectral behavior is realizable by a strong matrix on a graph, then it often remains realizable for any graph containing it as a minor. In this talk, we will survey these connections and present new results on strong properties for discrete Schrödinger operators.